There were at least two earlier MO questions about ideal pocket billiards. (Ideal: frictionless, perfectly elastic collisions.)
Perfectly centered break of a perfectly aligned pool ball rack.
Does there exist a shot in ideal pocket billiards?.
Answered Not always by George Lowther with a clever example:
My question is a variation on whether there is always a shot. George Lowther's example involved touching or nearly touching balls, and collinear balls, and forces the cue ball into a pocket--a "scratch." Here I ask for more generic situations:
Q. With the initial ball centers in general position (no three collinear), is there an $\epsilon > 0$ such that, if initially every pair of balls is separated by at least $\epsilon$, there always exists a shot that sinks at least one ball and avoids a scratch?
My guess is Yes, perhaps with $\epsilon = \frac{1}{2}$ (and unit-radius balls). And maybe there is always a shot that sinks all $15$ balls?
$\mathfrak{Merry}$ $\mathfrak{Christmas!}$
